The invention relates to a method for applying a metallization in accordance with a pattern of a system of mutually connected electrical conductors for transporting electrical charge carriers on at least one of the outer surfaces of a photovoltaic element, wherein the conductors display a determined series resistance and cover a determined fraction of at least the one surface, which method comprises the steps of
(i) providing said photovoltaic element, at least one of the outer surfaces of which is adapted for applying of a metallization thereto, and
(ii) applying a metallization in accordance with a determined pattern on the relevant surface.
A photovoltaic element (solar cell) generally comprises a stacked structure of at least one layer of semiconductor material of the p-type and a layer of semiconductor material of the n-type for generating electrical charge carriers under the influence of photons incident on the interface of these layers; a pattern of a system of mutually connected conductors, hereinafter referred to as metallization pattern, forms an essential part of such an element. A metallization pattern is usually applied to the front and rear surface of a solar cell, for instance using screen-printing techniques, in order to make electrical contacts with this cell. Applied metallization is usually non-transparent and causes shadow losses in a solar cell on the side receiving the sunlight (further designated as front side).
The design of a metallization pattern applied by means of screen-printing to the front side of a solar cell is generally the result of a compromise between shadow losses caused by this metallization and the electrical series resistance of the metallization pattern: the lower the resistance, the greater the total surface area of the pattern and the shadow losses caused thereby. On the rear side of a solar cell shadow losses generally do not represent a problem, completely covering the rear side with a metallization layer is however not strictly necessary for proper functioning of a solar cell, while the production costs of such a full layer are generally higher than those of a metallization pattern which does not completely cover the rear side.
A metallization pattern usually comprises a web of thin lines, referred to as fingers. These lines have to be thin so that they can be arranged close to each other in order to thus reduce losses resulting from the resistance in the material of the photovoltaic element (the emitter resistance). The fingers themselves generally have insufficient electrical conductivity to conduct a photo current to the edge of the photovoltaic element without considerable losses. In order to reduce losses in the fingers, contact strips, so-called tabs, are usually arranged over the fingers which increase the total conductivity and also serve to connect a solar cell to another solar cell.
A web of fingers can in principle have any topology and dimensions, so that it is particularly difficult to find a generally optimal pattern. Optimizing of a metallization pattern therefore generally takes place for a group of patterns within a determined topology.
In Solid-State Electronics 37 (1994) Jan. No. 1. p. 220-222, a method has been developed which permits optimum grid-line patterns to be determined based on the calculation of a single parameter, the carrier mean path. The solar cell with the minimum value of this parameter has the most effective grid pattern. The method is specially indicated in the case of comparing solar cells with grid patterns of different shapes and the same shadowing factor and total grid-line lengths.
It is a drawback of this known method that the direction of the fingers is fixed by the choice of the peculiar pattern and is not chosen optimally. Another drawback of the known optimization method is that it is limited to solar cells with a particular, generally rectangular geometry.
From JP-A-06053531, i.e. the corresponding Patent Abstract of Japan, it is known to enhance the photoelectric conversion efficiency of a photoelectric converter by suppressing Joule heat and optical loss, by providing a solar cell in which subelectrodes are thinnest at a part remotest from a main electrode and become thicker toward this main electrode.
The object of the invention is to provide a method for applying a metallization pattern optimized for resistance losses and shadow losses, irrespective of the geometry of the relevant photovoltaic element.
This object is achieved, and other advantages gained, with a method of the type stated in the preamble, wherein according to the invention the metallization is applied in the second step (ii) in accordance with an optimized pattern, the geometry of which is defined such that the electrical power of this element is maximal as a function of this geometry.
The invention is based on the surprising insight that it is possible to express a known metallization pattern of a system of mutually connected electrical conductors on a solar cell in an at least partially transparent metallization which completely covers this solar cell, wherein the same shadow and resistance losses occur as in this known metallization pattern.
It has now been found that it is conversely possible to construct a system of mutually connected electrical conductors from an at least partially transparent metallization which completely covers a solar cell, wherein the same shadow and resistance losses occur as in this at least partially transparent metallization.
In an embodiment of the method according to the invention the optimized pattern for the purpose of applying a metallization pattern causing shadow and resistance losses on a front surface of the photovoltaic element adapted to receive incident light is obtained by determining the thickness as a function of the location, and therefore determining the progression of the thickness, on said front surface of a light-transmitting metallization layer which completely covers this front surface and allows through a fraction of a quantity of incident light corresponding with said degree of covering, in a manner such that the sum of the shadow and resistance losses which would occur in this element in operative state is minimal and deriving from this thickness as a function of location the geometry of a pattern of a system of mutually connected non-light-transmitting electrical conductors.
In another embodiment of the method according to the invention the optimized pattern for the purpose of applying a metallization pattern causing resistance losses on a rear surface of the photovoltaic element lying opposite a front surface adapted to receive incident light is obtained by determining the thickness as a function of the location on said rear surface of a light-transmitting metallization layer which completely covers this rear surface and allows through a fraction of a quantity of incident light corresponding with said degree of covering in a manner such that the sum of the resistance losses and a measure expressed in loss of efficiency for the quantity of material for the metallization which would occur in this element in operative state is minimal, and deriving from this thickness as a function of the location the geometry of a pattern of a system of mutually connected non-light-transmitting electrical conductors.
In an embodiment of a method according to the invention the thickness as a function of the location of the respective metallization layer is generally determined for a case in which the metallization layer which would completely cover the respective surface has an isotropic electrical conductivity.
In a particular case, for instance when optimizing a metallization pattern with a pattern of lines which conducts the electrical current generated in the relevant solar cell in a determined direction, the thickness as a function of the location of the respective metallization is determined for a case in which the metallization layer which would completely cover the respective surface has an anisotropic electrical conductivity.
In a following embodiment of the method the optimized pattern is obtained by determining the thickness of the respective light-transmitting metallization layer as a function of the location on the respective surface, in a manner such that the electrical power of this element is also maximal as a function of the electrical resistance in the transition between the surface of the element and the metallization.
In yet another embodiment the optimized pattern is obtained by determining the thickness of the respective light-transmitting metallization layer as a function of the location on the respective surface, in a manner such that the electrical power of this element is also maximal as a function of the surface resistance of the element.
In yet another following embodiment the optimized pattern is obtained by determining the thickness of the respective light-transmitting metallization layer as a function of the location on the respective surface, in a manner such that the electrical power of this element is also maximal as a function of the resistivity of the material of the metallization.
Determining of the thickness of the metallization for a number of points distributed over a front surface in a manner such that the sum of the occurring shadow and resistance losses which would occur in this element in operative state in the presence of a metallization layer completely covering this front surface is minimal, is for instance performed as follows.
If in a point represented by coordinates (x, y), on the surface of the front side of a photovoltaic element the local current density and the voltage at the maximum power generated by this element are represented by respectively Jmp and Vmp, the total generated power Pg in the absence of shadow losses caused by a metallization layer is then given by the equation
Pgxe2x88x92∫sJmpVmpdxdyxe2x80x83xe2x80x83(1)
in which S is the surface of the element.
The maximum power represented by equation (1) is in practice lower as a result of the occurring shadow and resistance losses. On the front side of the element the current density is reduced as a result of shadow effects caused by the necessarily present metallization:
Jxe2x80x2mp(x,y)=Jmp(1xe2x88x92ps(x,y))xe2x80x83xe2x80x83(2)
in which ps (x,y) represents the shadow fraction which is given by                                           p            s                    ⁡                      (                          x              ,              y                        )                          =                              d            ⁡                          (                              x                ,                y                            )                                            d            0                                              (        3        )            
in which d(x,y) is the thickness as a function of the location of an at least partially transparent metallization layer and do the thickness of a conductor in a system of mutually connected conductors.
From ps (x,y) a location-dependent effective surface resistance xcfx81sm (x,y) can be determined:                                           ρ            sm                    ⁡                      (                          x              ,              y                        )                          =                              ρ                          sm              ,              0                                                          p              s                        ⁡                          (                              x                ,                y                            )                                                          (        4        )            
in which xcfx81sm,0 is the surface resistance of the lines forming the metallization pattern.
The ohmic losses in the metallization are determined by solving a partial differential equation for the voltage distribution V(x,y):                                           ∇            →                    ⁢                      ·                          (                                                1                                                            ρ                      sm                                        ⁡                                          (                                              x                        ,                        y                                            )                                                                      ⁢                                                      ∇                    →                                    ⁢                                      V                    ⁡                                          (                                              x                        ,                        y                                            )                                                                                  )                                      =                  -                                    J              mp              xe2x80x2                        ⁡                          (                              x                ,                y                            )                                                          (        5        )            
The pattern of the electrical currents resulting from the solution of this differential equation (5) is the pattern in which the least ohmic dissipation occurs. The differential equation must be solved with boundary conditions, given by the voltage on the connection points for external contacts of the relevant solar cell.
Equation (5) enables calculation of the voltage distribution V(x,y) and the direction of the current density in the metallization, from the geometry of the solar cell, the surface resistance xcfx81sm(x,y) and the local current density Jxe2x80x2mp(x,y). The power dissipated in the metallization is given by:                               p          m                =                              ∫            S                    ⁢                                    1                                                ρ                  sm                                ⁡                                  (                                      x                    ,                    y                                    )                                                      ⁢                                          (                                                      ∇                    →                                    ⁢                                      V                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      )                            2                        ⁢                          ⅆ              x                        ⁢                          ⅆ              y                                                          (        6        )            
Ohmic losses likewise occur in the transition between element surface (the emitter) and the metallization. The contact between emitter and metallization is represented by an a effective contact resistance xcfx81ce which depends on the finger width of the relevant metallization and the surface resistance under the emitter. The power dissipated as a result of the contact resistance is represented by:                               p          c                =                              ρ            ce                    ⁢                                    ∫              S                        ⁢                                                                                                      J                      mp                      xe2x80x2                                        ⁡                                          (                                              x                        ,                        y                                            )                                                        2                                                                      p                    s                                    ⁡                                      (                                          x                      ,                      y                                        )                                                              ⁢                              ⅆ                x                            ⁢                              ⅆ                y                                                                        (        7        )            
in which ps(x,y) represents a reduction factor of the contact surface which is a result of partial metallization.
A further cause of ohmic losses is formed by the emitter surface resistance, xcfx81e. It can be derived that the power Pe per unit of area dissipated as a result hereof for a metallization pattern of parallel fingers with a width w is given by:                               p          e                =                              ∫            S                    ⁢                                                    w                2                            12                        ⁢                          (                                                1                  -                                                            p                      s                                        ⁡                                          (                                              x                        ,                        y                                            )                                                                                                                                  p                      s                                        ⁡                                          (                                              x                        ,                        y                                            )                                                        2                                            )                        ⁢                          ρ              e                        ⁢                                                            J                  mp                  xe2x80x2                                ⁡                                  (                                      x                    ,                    y                                    )                                            2                        ⁢                          ⅆ              x                        ⁢                          ⅆ              y                                                          (        8        )            
No power is of course generated in the part of the surface of a solar cell covered by the metallization pattern. The quantity of power not generated as a result of covering the solar cell surface, referred to as the shadow loss Ps, is represented by:
Ps=∫sPs(x,y)JmpVmpdxdyxe2x80x83xe2x80x83(9)
Shadow losses generally play no significant part on the rear side of a solar cell. It is however important to limit the quantity of material for a metallization layer on the rear side. It can be derived that a quantity of material to be optimized can be expressed in terms of a shadow loss, as follows:
Ps=∫sPs(x,y)JmpVmpFcdxdyxe2x80x83xe2x80x83(10)
Fc is here a dimensionless parameter which relates the quantity of material to an efficiency loss.
The total loss of power Pt occurring in a solar cell can thus be expressed in terms of a shadow loss and occurring ohmic losses:
Pt=Ps+Pm+Pc+Pexe2x80x83xe2x80x83(11)
According to the invention an optimal location-dependent shadow fraction ps(x,y) is determined. Losses Pc and Pe respectively resulting from occurring contact resistance and emitter surface resistance as well as the shadow loss Ps follow directly from this shadow fraction using equations (7) and (8). From the shadow fraction the surface resistance xcfx81sm (x,y) can be calculated using equation (4), and the current density Jxe2x80x2mp can be calculated using equation (2). The partial differential equation (5) can be solved using the surface resistance xcfx81sm(x,y). Once this equation is solved, the ohmic losses in the metallization can be determined. The shadow fraction ps(x,y) must be chosen such that the total loss Pt is minimal.
The optimization is performed as follows. A main program, written in C, addresses a subroutine written in Fortran which finds a local optimum of a continuous and differentiable function of a number of parameters which can be freely chosen. The subroutine has upper and lower limits for the parameters. The parameters are the shadow fractions ps(xi,yi) at a set of points (xi,yi) to be defined by the user, designated hereinafter as coarse mesh.
The function is in this case a relatively complicated construction which calculates the total loss Pt from the shadow fractions on the relevant coarse mesh. For a random point (x, y), ps(x,y) is defined by means of interpolation from the values ps(xi,yi). The partial differential equation (5) is solved with a finite element method (FEM).
The integrands for the contact resistance loss (7), the emitter surface resistance loss (8) and the generated power are evaluated on the fine-mesh network of the finite element method, wherein for each integrand an interpolant is created which is then integrated.
Once the optimal shadow fraction ps(x,y) is determined, a translation can be performed to an equivalent metallization pattern. The orientation of the fingers is given by the field strength {right arrow over (∇)}V(x,y). The average shadow fraction can be translated to a position-dependent finger distance s(x,y):                               s          ⁡                      (                          x              ,              y                        )                          =                  W                                    p              s                        ⁡                          (                              x                ,                y                            )                                                          (        12        )            
It has been found that in a solar cell provided with a screen-printed metallization pattern according to the invention, the metallization losses, which amount to about 12 to 13% in a pattern according to the prior art, are reduced to about 7%.